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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X3849 X9,16,10,17 X17,20,18,5 X11,19,12,18 X19,11,20,10 X13,2,14,3

Gauss Code:

{{1, 10, -5, -3}, {-4, -1, 2, 5, -6, 9, -8, 4, -10, -2, 3, 6, -7, 8, -9, 7}}

Jones Polynomial:

- q-17/2 + q-15/2 - q-13/2 + 2q-11/2 - 2q-9/2 + q-7/2 - 2q-5/2 + q-3/2 - q-1/2

A2 (sl(3)) Invariant:

q-30 + q-26 + q-24 - q-20 - 2q-18 + 2q-12 + 2q-10 + 2q-8 + q-6 + q-4 + q-2

HOMFLY-PT Polynomial:

- 2a3z-1 - 6a3z - 5a3z3 - a3z5 + 4a5z-1 + 9a5z + 11a5z3 + 6a5z5 + a5z7 - 3a7z-1 - 6a7z - 5a7z3 - a7z5 + a9z-1 + a9z

Kauffman Polynomial:

- 2a3z-1 + 8a3z - 11a3z3 + 6a3z5 - a3z7 + 2a4 - 2a4z2 - 5a4z4 + 5a4z6 - a4z8 - 4a5z-1 + 17a5z - 28a5z3 + 17a5z5 - 3a5z7 + 3a6 - 5a6z2 - a6z4 + 4a6z6 - a6z8 - 3a7z-1 + 12a7z - 18a7z3 + 11a7z5 - 2a7z7 + 3a8 - 4a8z2 + 4a8z4 - a8z6 - a9z-1 + 2a9z - a9z3 + a10 - a10z2 - a11z

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                 1 j = -2                   j = -4             2 1   j = -6         1 1 1     j = -8         2 1       j = -10     1 2 1         j = -12   1 2 2           j = -14   1 1             j = -16 1 1               j = -18 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 10]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 10]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 12, 6, 13], > X[3, 8, 4, 9], X[9, 16, 10, 17], X[17, 20, 18, 5], X[11, 19, 12, 18], > X[19, 11, 20, 10], X[13, 2, 14, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -5, -3}, {-4, -1, 2, 5, -6, 9, -8, 4, -10, -2, 3, 6, -7, 8, > -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) -(15/2) -(13/2) 2 2 -(7/2) 2 -(3/2) -q + q - q + ----- - ---- + q - ---- + q - 11/2 9/2 5/2 q q q 1 > ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -26 -24 -20 2 2 2 2 -6 -4 -2 q + q + q - q - --- + --- + --- + -- + q + q + q 18 12 10 8 q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 10]][a, z]
Out[8]=	3 5 7 9 -2 a 4 a 3 a a 3 5 7 9 3 3 ----- + ---- - ---- + -- - 6 a z + 9 a z - 6 a z + a z - 5 a z + z z z z 5 3 7 3 3 5 5 5 7 5 5 7 > 11 a z - 5 a z - a z + 6 a z - a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 10]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 2 a 4 a 3 a a 3 5 2 a + 3 a + 3 a + a - ---- - ---- - ---- - -- + 8 a z + 17 a z + z z z z 7 9 11 4 2 6 2 8 2 10 2 > 12 a z + 2 a z - a z - 2 a z - 5 a z - 4 a z - a z - 3 3 5 3 7 3 9 3 4 4 6 4 8 4 > 11 a z - 28 a z - 18 a z - a z - 5 a z - a z + 4 a z + 3 5 5 5 7 5 4 6 6 6 8 6 3 7 > 6 a z + 17 a z + 11 a z + 5 a z + 4 a z - a z - a z - 5 7 7 7 4 8 6 8 > 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 2 1 1 1 1 1 1 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 18 6 16 6 16 5 14 5 12 5 14 4 12 4 q q t q t q t q t q t q t q t 1 2 2 1 2 1 1 1 t 2 > ------ + ------ + ------ + ------ + ----- + ----- + ---- + ---- + -- + t 10 4 12 3 10 3 10 2 8 2 6 2 8 6 4 q t q t q t q t q t q t q t q t q

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